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Title:Basic stochastic transmission models and their inference

Abstract: The current survey paper concerns stochastic mathematical models for the
spread of infectious diseases. It starts with the simplest setting of a
homogeneous population in which a transmittable disease spreads during a short
outbreak. Assuming a large population some important features are presented:
branching process approximation, basic reproduction number $R_0$, and final
size of an outbreak. Some extensions towards realism are then discussed: models
for endemicity, various heterogeneities, and prior immmunity. The focus is then
shifted to statistical inference. What can be estimated for these models for
various levels of detailed data and with what precision? The paper ends by
describing how the inference results may be used for determining successful
vaccination strategies. This paper will appear as a chapter of a forthcoming
book entitled \emph{Handbook of Infectious Disease Epidemiology}.